Sturm's Theorem (redirected from Sturmian Sequence)

Sturm's theorem [′stərmz ‚thir·əm]

(mathematics)

This gives a method to determine the number of real roots of a polynomialp(x) which lie between two given values ofx; the Sturm sequence ofp(x) provides the necessary information.

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Sturm’s Theorem 

a theorem that provides a basis for finding nonoverlapping intervals such that each contains one real root of a given algebraic polynomial with real coefficients. The theorem was given in 1829 by J. C. F. Sturm.

For any polynomial f(x) without multiple roots, there exists a system of polynomials

f(x) = f₀(x), f₁(x),...,f_s(x)

for which the following conditions are fulfilled: (1) f_k(x) and f_k+1(x), k = 0, 1,..., s – 1, do not have common roots; (2) the polynomial f_s (x) has no real roots; (3) it follows from f_k (α) = 0, 1 ≤ k ≤ s –1 that f_k–1 (α)f_k+1 (α) < 0; and (4) it follows from f(α) = 0 that the product f(x)f₁(x) is increasing at the point α. Let w(c) be the number of changes of sign in the system

f(c), f₁(c),...,f_s(c)

If the real numbers a and b (a < b) are not roots of the polynomial f(x), then the difference w(a) – w(b) is nonnegative and equal to the number of real roots of the polynomial f(x) that lie between a and b. Thus, the number line may be divided into intervals each of which contains one real root of the polynomial f(x).